Differential operations dl= fdr+ Or de or dv=r2 sing dr de dp (D.103) ds=rasin g de do (D.104) dSe=rsin6drdφ dSo=drdo (D.106) dr Vf rar(Fr)Ursine ag(sin8F0)+aFg (D.108) rsing d r2 sin e 品a Fr r Fe 1 af =-2(+0+如面+m)+ cos e +a7Fe-r2sin26Fe-2a0+ dFr cos a Fe Separation of the Helmholtz equation ay(r,6,φ) ay(r,6,φ) sin 0 a2 y(r,6,φ)=0 y(r,6,φ)=R(r)e(6)Φ(φ) D.11 (D.114) n+ 1-n2 d-e(n de(n dn2 ≤n≤ @2001 by CRC Press LLCDifferential operations dl = rˆ dr + θˆr dθ + φˆr sin θ dφ (D.102) dV = r 2 sin θ dr dθ dφ (D.103) d Sr = r 2 sin θ dθ dφ (D.104) d Sθ = r sin θ dr dφ (D.105) d Sφ = r dr dθ (D.106) ∇ f = rˆ ∂ f ∂r + θˆ 1 r ∂ f ∂θ + φˆ 1 r sin θ ∂ f ∂φ (D.107) ∇ · F = 1 r 2 ∂ ∂r r 2Fr + 1 r sin θ ∂ ∂θ (sin θ Fθ ) + 1 r sin θ ∂Fφ ∂φ (D.108) ∇ × F = 1 r 2 sin θ rˆ rθˆ r sin θφˆ ∂ ∂r ∂ ∂θ ∂ ∂φ Fr r Fθ r sin θ Fφ (D.109) ∇2 f = 1 r 2 ∂ ∂r  r 2 ∂ f ∂r  + 1 r 2 sin θ ∂ ∂θ  sin θ ∂ f ∂θ  + 1 r 2 sin2 θ ∂2 f ∂φ2 (D.110) ∇2 F = rˆ  ∇2Fr − 2 r 2  Fr + cos θ sin θ Fθ + 1 sin θ ∂Fφ ∂φ + ∂Fθ ∂θ  + + θˆ  ∇2Fθ − 1 r 2  1 sin2 θ Fθ − 2 ∂Fr ∂θ + 2 cos θ sin2 θ ∂Fφ ∂φ  + + φˆ  ∇2Fφ − 1 r 2  1 sin2 θ Fφ − 2 1 sin θ ∂Fr ∂φ − 2 cos θ sin2 θ ∂Fθ ∂φ  (D.111) Separation of the Helmholtz equation 1 r 2 ∂ ∂r  r 2 ∂ψ(r,θ,φ) ∂r  + 1 r 2 sin θ ∂ ∂θ  sin θ ∂ψ(r,θ,φ) ∂θ  + + 1 r 2 sin2 θ ∂2ψ(r,θ,φ) ∂φ2 + k2 ψ(r,θ,φ) = 0 (D.112) ψ(r,θ,φ) = R(r)(θ)(φ) (D.113) η = cos θ (D.114) 1 R(r) d dr  r 2 d R(r) dr  + k2 r 2 = n(n + 1) (D.115) (1 − η2 ) d2(η) dη2 − 2η d(η) dη +  n(n + 1) − µ2 1 − η2  (η) = 0, −1 ≤ η ≤ 1 (D.116)